Understanding Vertical Tangent Lines in Mathematics | Definition, Examples, and Applications

Vertical tangent line definition

A vertical tangent line is a line that is perpendicular to the x-axis and touches a curve or a graph at a specific point where the slope of the curve is undefined

A vertical tangent line is a line that is perpendicular to the x-axis and touches a curve or a graph at a specific point where the slope of the curve is undefined. In other words, it is a line that is perfectly vertical and touches the curve but does not cross or intersect it.

To understand this concept better, let’s consider an example. Consider the graph of a function f(x) on a coordinate plane. If at a specific point P on the graph, the slope of the curve (represented by the tangent line at that point) becomes infinite, then the tangent line at that point is said to be a vertical tangent line. This occurs when the curve abruptly changes direction, causing the slope to become undefined.

Mathematically, if the equation of the curve is y = f(x), then the slope of the curve at the point (a, f(a)) is given by the derivative of the function f(x) evaluated at x = a, denoted as f'(a). When f'(a) is undefined or tends to infinity, a vertical tangent line occurs at (a, f(a)).

For example, consider the function f(x) = √x. This function has a graph that starts from the origin and continues upwards. At the point (0,0), when we try to find the slope of the tangent line using the derivative, we get the expression f'(0) = 1/(2√0). Since division by zero is undefined, the slope becomes infinite. Hence, the line passing through (0,0) is a vertical tangent line.

It is important to note that not all curves have vertical tangent lines. Only functions that display sudden changes in direction or sharp corners will have this phenomenon. In most cases, when the slope of the curve remains finite at a specific point, the tangent line will be inclined, rather than vertical.

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